HGAN-SDEs: Learning Neural Stochastic Differential Equations with Hermite-Guided Adversarial Training

• URL: http://arxiv.org/abs/2512.20272v1
• Date: Tue, 23 Dec 2025 11:25:22 GMT
• Title: HGAN-SDEs: Learning Neural Stochastic Differential Equations with Hermite-Guided Adversarial Training
• Authors: Yuanjian Xu, Yuan Shuai, Jianing Hao, Guang Zhang,
• Abstract summary: We introduce HGAN-SDEs, a novel GAN-based framework that leverages Neural Hermite functions to construct a structured and efficient discriminator.<n>HGAN-SDEs achieve superior sample quality and learning efficiency compared to existing generative models for SDEs.
• Score: 3.4515388499147654
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: Neural Stochastic Differential Equations (Neural SDEs) provide a principled framework for modeling continuous-time stochastic processes and have been widely adopted in fields ranging from physics to finance. Recent advances suggest that Generative Adversarial Networks (GANs) offer a promising solution to learning the complex path distributions induced by SDEs. However, a critical bottleneck lies in designing a discriminator that faithfully captures temporal dependencies while remaining computationally efficient. Prior works have explored Neural Controlled Differential Equations (CDEs) as discriminators due to their ability to model continuous-time dynamics, but such architectures suffer from high computational costs and exacerbate the instability of adversarial training. To address these limitations, we introduce HGAN-SDEs, a novel GAN-based framework that leverages Neural Hermite functions to construct a structured and efficient discriminator. Hermite functions provide an expressive yet lightweight basis for approximating path-level dynamics, enabling both reduced runtime complexity and improved training stability. We establish the universal approximation property of our framework for a broad class of SDE-driven distributions and theoretically characterize its convergence behavior. Extensive empirical evaluations on synthetic and real-world systems demonstrate that HGAN-SDEs achieve superior sample quality and learning efficiency compared to existing generative models for SDEs

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